Approximations and fixed points for condensing non-self-maps defined on a sphere

Abstract

In this paper, we investigate the validity of an interesting theorem of Ky Fan [Theorem 2, Math. Z. 112 (1969), 234-240] defined on a sphere (the boundary of a closed ball) in an infinite-dimensional Banach space. We will prove that it is true for a continuous condensing map with suitable conditions posed. As applications of our theorem, some fixed point theorems of continuous condensing non-self-maps defined on a sphere are derived. Our results generalize some results of R. Nussbaum [10] and P. Massatt [8]. Most fixed point theorems in Banach spaces deal with some classes of maps defined on a compact convex (or star-shaped) subset, a closed bound convex (or star-shaped) subset, or a closed bound subset with nonempty interior. What about the domain of a function which is neither of the above cases? The simplest interesting case would be a sphere (the boundary of a closed ball). It is clear that a continuous self-map defined on a sphere may not have fixed points; for example, a rotation of a sphere (or circle) in a plane. Nussbaum [10] proved that a continuous k-set-contractive map from a sphere into a sphere has a fixed point, if the dimension of the Banach space is infinite. Recently, Massatt [8] generalized this result to continuous condensing maps. For definitions of k-setcontractive and condensing maps, see for example [9 or7] . Generalizations of fixed point theory from self-maps to non-self-maps has been a very active topic in nonlinear functional analysis in the past two decades. Does a continuous condensing non-self-map defined on a sphere have a fixed point? Under which conditions? On the other hand, Fan [3] proved the following interesting theorem: Let K be a nonempty compact convex subset of a normed linear space X. Let f be a continuous map from K into X; then there exists a point u in K such that Ilu f(u)II = d(f (u), K) . The author [5] proved that it is true for a continuous condensing map defined on a closed ball of a Banach space. Is this still true for a continuous condensing map defined on a sphere of an infinite-dimensional Banach space? Under which conditions? Received by the editors March 23, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 47H10; Secondary 41A50. ? 1989 American Mathematical Society 0002-9939/89 $1.00 + $ 25 per page